I have to say I am a believer of hardening up and am not a fan of excessive PC behaviour. Some of the things that JC has done was teetering on the edge of taste, but you go to ANY comedy show, and you will get things 10x worse. I don't like the BBC telling me what I should and should not be offended by.
The rhyme (heck I was taught that nursery rhyme as a child and only stopped using it when I realised what I was saying), the truckers joke, the Mexican jokes etc... I mean really? I've seen more offensive humour in an Amish town in America... What's happened to the British stiff upper lip, thick skin? Heck I'm a Johnny Foreigner and should be more sensitive to this sort of thing but I find it a riot. There is offence with intent (i.e. truly racist people, with vitriolic and aggressive xenophobic views) and then there's the contextual side of certain words and sentences which aren't meant to offend, but highlight something for the sake of humour.
However, that said, these are based on my own views/values. I do understand why these guidelines are needed in society/companies, and from a purely black and white point of view, they have made the right decision to fire him - that I cannot disagree with. It's just a shame it's come to this.